Transcript of Heteronuclear Molecules from a Quantum Degenerate Fermi ...
Heteronuclear Molecules from a Quantum Degenerate Fermi-Fermi
Mixtureder Ludwig–Maximilians–Universitat
Zweitgutachter: Prof. Dr. Wilhelm Zwerger
Tag der mundlichen Prufung: 16. September 2009
Meinen Eltern
Abstract
This thesis reports on experiments with quantum degenerate atomic
mixtures and molecular gases at ultracold temperatures. The work
describes the first realization of a quantum de- generate
Fermi-Fermi mixture of two different species with unequal masses.
In addition, the first quantum degenerate three-species mixture is
realized. Furthermore, the first heteronu- clear bosonic molecules
with temperatures close to quantum degeneracy are created from a
two-species mixture.
Within this work a new and very versatile experimental platform to
study quantum de- generate two-species Fermi mixtures is presented.
The experimental concept relies on sym- pathetic cooling of two
fermionic species (40K and 6Li) by a large bosonic gas (87Rb). It
is shown that large atom numbers and reliable operation are
guaranteed by careful choice of the experimental components and
parameters, which is essential to deal with the complexity of such
an experimental platform with three atomic species.
The first important milestone towards quantum degeneracy is
realized by simultaneous trapping 87Rb, 40K and 6Li in a
magneto-optical trap. This marks the first realization of
magneto-optical trapping of two fermionic species and also of three
species. To achieve quan- tum degeneracy, different experimental
challenges and difficulties of the three species with very
different initial temperatures, scattering cross sections and
masses were overcome. A combined compressed MOT and temporal dark
MOT phase, a careful state cleaning process during the cooling
process, species selective evaporative cooling of rubidium and
removing of high energetic lithium atoms turn out to be essential
for the successful realization of a quantum degenerate
Fermi-Fermi-Bose mixture. Furthermore, at the end of the
sympathetic cooling process, the cooling efficiency of lithium is
improved by catalytic cooling through potassium.
A quantum degenerate Fermi-Fermi mixture can be realized in the
strongly interacting regime at a s-wave Feshbach resonance. By an
adiabatic magnetic field sweep across the Fesh- bach resonance
ultracold bosonic heteronuclear molecules are created with high
efficiencies of up to 50%. The associated number of molecules are
up to 4 × 104 molecules, which is the largest number of
heteronuclear molecules produced so far. Furthermore, close to
resonance an increased molecular lifetime of more than 100ms is
observed. Therefore, this is the first system that may be used to
explore many-body physics of a heteronuclear mixture in the
strongly interacting regime across a Feshbach resonance.
The heteronuclear Fermi mixture adds the mass ratio and the
different internal structure as new parameters to quantum
degenerate Fermi gases. The molecular cloud being close to quantum
degeneracy marks the first starting point for the realization of a
heteronuclear molecular BEC and to investigate the BEC-BCS
crossover with unequal masses. Moreover, the molecules can be
transferred into the rovibrational ground state to create a polar
BEC with an anisotropic, long-range interaction.
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Zusammenfassung
Im Rahmen dieser Arbeit wird eine neue, vielseitig einsetzbare
experimentelle Plattform zum Studium von quantenenarteten
fermionischen Mischungen aus unterschiedlichen Atom- sorten
prasentiert. Das experimentelle Konzept beruht auf der
sympathetischen Kuhlung von zwei fermionischen Atomsorten (40K und
6Li) mit einem großen bosonischen Gas (87Rb). Es wird gezeigt, daß
große Atomzahlen und ein zuverlassiges Arbeiten durch eine
sorgfaltige Wahl der experimentellen Komponenten und Parameter
erfullt werden kann. Dies ist essenziell, um die Komplexitat eines
solchen Experiments mit drei Spezies zu bewaltigen.
Der erste wichtige Meilenstein zur Quantenentartung ist durch das
simultane Fangen von 87Rb, 40K und 6Li in einer magneto-optischen
Falle realisiert. Dies stellt das erste magneto- optische Fangen
von zwei fermionischen als auch von drei Atomsorten dar. Zur
Quantenentar- tung wurden unterschiedlichste experimentelle
Herausforderungen mit den drei Atomsorten, die verschiedene
Anfangstemperaturen, Streuquerschnitte und Massen aufweisen,
uberwun- den. Fur eine erfolgreiche Herstellung einer
quantenentarteten Fermi-Fermi-Bose-Mischung sind eine Kombination
aus komprimierter MOT und zeitlicher Dunkel-MOT Phase, eine sorg-
faltige Zustandsreinigung wahrend der Kuhlphase, speziesselektive
Verdampfungskuhlung von Rubidium und Entfernung von
hochenergetischen Lithium Atomen entscheidend. Ferner wird die
Kuhleffizienz von Lithium durch katalytisches Kuhlen durch Kalium
verstarkt.
Eine quantenentartete Fermi-Fermi-Mischung kann im
starkwechselwirkendem Regime an einer s-Wellen-Feshbach-Resonanz
prapariert werden. Mit Hilfe einer adiabatischen Magnet-
feldveranderung uber die Feshbach-Resonanz konnen ultrakalte
bosonische Molekule mit ho- hen Effizienzen bis zu 50% hergestellt
werden. Die assoziierten Molekulzahlen sind bis zu 4× 104, was die
derzeit großten heteronuklearen Molekulzahlen darstellt.
Daruberhinaus wei- sen die Molekule eine ansteigende Lebensdauer
nahe der Resonanz von mehr als 100ms auf. Demzufolge ist es das
erste System, mit dem Vielteilchenphysik einer heteronuklearen Mi-
schung im starkwechselwirkenden Regime an einer Feshbach-Resonanz
studiert werden kann.
Die heteronukleare Fermi-Mischung erweitert die quantenenarteten
Fermi-Gase um das Massenverhaltnis und die unterschiedliche interne
Struktur. Das molekulare Gas, welches nahe an der Quantenentartung
ist, markiert den ersten Startpunkt hin zu einem heteronuklearen
Molekul-BEC und der Untersuchung des BEC-BCS-Ubergangs mit
unterschiedlichen Massen. Daruberhinaus konnen die Molekule in den
rovibrationellen Grundzustand uberfuhrt werden, um ein polares BEC
mit einer anisotropen, langreichweitigen Wechselwirkung
herzustellen.
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Contents
1.3 Many-body physics and the BEC-BCS crossover . . . . . . . . . .
. . . . . . . 4
1.4 Ultracold chemistry . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 5
1.5 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 7
2.1.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 11
2.1.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 13
2.2.3 The asymptotic bound state model . . . . . . . . . . . . . .
. . . . . . . 18
2.2.4 Classification of Feshbach resonances . . . . . . . . . . . .
. . . . . . . . 20
2.2.5 Fermi-Fermi mixtures . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 22
3 Experimental Setup 25
3.1 Experimental concept . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 25
3.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 26
3.3 Atomic sources . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 28
3.3.2 Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 28
3.4 Laser systems . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 29
3.4.1 Energy levels . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 29
3.4.5 Optical system for MOT, detection and optical pumping . . . .
. . . . . 33
3.4.6 Optical dipole trap . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 33
3.5 Absorption imaging . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 36
3.6 Magnetic trapping . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 38
3.10 Experimental Control . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 47
4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 49
4.2.1 Single MOTs . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 50
4.2.2 Triple MOT . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 51
4.2.3 Dispenser currents . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 51
4.2.4 Light-assisted collisions . . . . . . . . . . . . . . . . . .
. . . . . . . . . 52
4.2.5 Optical molasses . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 53
5 Quantum degeneracy 55
5.1.1 cMOT and dMOT . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 55
5.1.2 State preparation . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 57
5.1.3 Magnetic transport . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 58
5.1.4 QUIC trap . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 59
5.2.1 Evaporative cooling . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 60
5.2.2 Sympathetic cooling . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 64
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 71
6.2 Magnetic field calibration . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 75
6.3 State preparation . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 77
6.4.1 Loss measurement . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 79
6.5.2 Reconversion process . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 81
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Contents
7 Conclusions and Outlook 87
A Natural constants and atomic sources 89 A.1 Natural constants . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89 A.2 Atomic properties . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 90
Bibliography 93
Danksagung 113
Introduction
Since the first realization of Bose-Einstein condensation in
ultracold, dilute gases 14 years ago (Anderson et al., 1995;
Bradley et al., 1995; Davis et al., 1995), followed by quantum
degen- eracy of fermions (DeMarco and Jin, 1999), the research
field on ultracold, dilute quantum gases has produced many exciting
experiments and has opened new prospects on atomic and many-body
physics, and also refocussed the interest on molecular
physics.
In the following, an overview on ultracold quantum gases and their
recent developments is given, first on bosons in Sec. 1.1 and then
on fermions in Sec. 1.2. After this their application to many-body
physics and ultracold chemistry are discussed in Sec. 1.3 and Sec.
1.4, respectively. Finally, Sec. 1.5 gives an outline of this
thesis.
1.1 Bose-Einstein condensation
Based on a work on photon statistics by Satyendra Nath Bose (Bose,
1924), Albert Einstein predicted Bose-Einstein condensation (BEC)
for an ideal Bose gas in the year 1925 (Einstein, 1925). This
phenomena of condensation is a consequence of quantum statistics
and postulates a macroscopic occupation of the single particle
ground state for temperatures below a critical value.
The connection between superflluid liquid 4He and Bose-Einstein
condensation was first suggested by Fritz London (London, 1938a,b).
However, due to large interparticle interactions in liquid helium,
the number of atoms in the single particle ground state is reduced
and only a phenomenological description of the superfluid phase is
possible. A very successful phenomenological model, given by L. D.
Landau, is the two-fluid model (Landau, 1949).
In dilute atomic gases, on the other side, interparticle
interactions are typically weak and allow to create pure
condensates. For typical atomic densities of 1013 − 1015 cm−3 the
timescale for the formation of molecules and for liquification is
much longer than the timescale needed for thermalization by
two-body collisions, which means that the two-body collision rate
dominates the three-body collision rate. This allows to achieve
Bose-Einstein condensation in a metastable gaseous phase for
temperatures far below the critical temperature.
The temperature needed for Bose-Einstein condensation in dilute
systems are on the order of hundreds of nanokelvin. This makes it
experimentally quite challenging to reach this regime. The first
important step towards ultralow temperatures was the proposal by
Hansch and Schawlow (1975) and the following experiments in the
1980s on laser cooling of neutral atomic gases (see Nobel lectures
by Chu (1998); Cohen-Tannoudji (1998); Phillips (1998)),
their
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1. Introduction
magnetic trapping (Migdall et al., 1985) and the realization of a
magneto-optical trap (MOT) (Raab et al., 1987). Here, the first
results for laser cooling were achieved with alkali atoms that have
certain advantages due to their relative simple energy-level
structure. Another key cooling technique is evaporative cooling
(Hess, 1986; Masuhara et al., 1988), which was developed in
experiments on spin polarized atomic hydrogen, before it was
applied to alkaline gases in magnetic or optical dipole traps
(Dalibard and Cohen-Tannoudji, 1985; Chu et al., 1986). This
technique allows to increase phase-space density by several orders
of magnitude. Current laser cooling techniques are limited in the
final phase-space density and up to now evaporative cooling is the
only used technique to achieve the quantum degenerate regime.
The first realization of Bose-Einstein condensation was achieved
with the alkali species 85Rb, 23Na and 7Li (Anderson et al., 1995;
Davis et al., 1995; Bradley et al., 1995). The list of condensed
species is still increasing and counts 14 different atomic species
now (Fried et al., 1998; Cornish et al., 2000; Robert et al., 2001;
Santos et al., 2001; Modugno et al., 2001; Weber et al., 2002;
Takasu et al., 2003; Griesmeier et al., 2005; Fukuhara et al.,
2007a,c; Sterr, 2009). Dilute Bose-Einstein condensations are prime
candidates to study quantum matter phenomena with precise control
over key parameters and offer a model system for many-body physics.
This has led to a boom of dilute quantum gases with almost one
hundred research groups worldwide. The first experiments on
Bose-Einstein condensation were focused on characteristic
properties like the interference of matter waves, elementary
excitations, the quantization of vortices and many more (see also
reviews on this subject by Dalfovo et al. (1999); Leggett (2001);
Pethick and Smith (2002); Pitaevskii and Stringari (2003)).
One interesting research direction that is relevant in the context
of this thesis is to tune the interparticle interaction by a
Feshbach resonance, which is originally a concept from nuclear
physics (Fano, 1935, 1961; Feshbach, 1958, 1962). A Feshbach
resonance occurs, if the collision energy of two atoms
energetically coincides with a molecular bound state. Such a
Feshbach resonance can be tuned precisely, for example, by an
external magnetic field (Tiesinga et al., 1993) or optically by
laser light (Fedichev et al., 1996a; Bohn and Julienne, 1997,
1999). On resonance the scattering length a has a singularity, a →
±∞, which in principle allows to vary a to arbitrary values. This
tunability makes Feshbach resonances a powerful tool in the field
of ultracold gases, where the first Feshbach resonances were
observed with bosonic atoms (Inouye et al., 1998; Courteille et
al., 1998; Roberts et al., 1998; Vuletic et al., 1999). However,
working with bosons close to a Feshbach resonance with large
interactions is quite challenging. In the vicinity of a Feshbach
resonance three-body losses are enhanced for bosonic atoms and
limits the molecular lifetime in a bulk gas significantly (Fedichev
et al., 1996b; Inouye et al., 1998; Vuletic et al., 1999; Stenger
et al., 1999; Roberts et al., 2000). Nevertheless, in a three-
dimensional optical lattice it is possible to create isolated
molecules on single lattice sites and to increase the molecular
lifetime significantly (Thalhammer et al., 2006; Volz et al.,
2006).
1.2 Quantum degenerate Fermi gases
After the successful creation of Bose-Einstein condensation in
dilute gases and exciting exper- iments on coherent, macroscopic
many-body states, the dramatic progress in the experimental methods
made quantum degeneracy in dilute fermionic gases possible. This
achievement al- lows to investigate differences in quantum
statistics between fermionic and bosonic systems. Furthermore,
fermions offer richer phenomena in the context of many-body
physics.
In contrast to bosons, non-interacting fermions have a smooth
crossover in their char-
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1.2 Quantum degenerate Fermi gases
acteristic thermodynamic properties, when they are cooled into
quantum degeneracy. The quantum degenerate regime for fermions is
generally defined for temperatures well below the Fermi temperature
TF = EF/kB, here EF is the Fermi energy. In 1999 the first quantum
de- generate Fermi gas was experimentally realized with 40K at JILA
(DeMarco and Jin, 1999). The list of degenerate Fermi gases is
currently enlarged by 6Li (Truscott et al., 2001; Schreck et al.,
2001), metastable 3He∗ (McNamara et al., 2006) and two earth alkali
species 171Yb and 173Yb (Fukuhara et al., 2007b).
From the experimental point of view, cooling fermions into quantum
degeneracy is more challenging than in the bosonic case. At
ultracold temperatures interparticle scattering is dominated by
s-wave collisions. But, due to the Pauli exclusion principle,
fermionic atoms of a one-component Fermi gas do not scatter with
each other, which makes evaporative cooling of a one-component
Fermi gas unfeasible. One strategy to circumvent this problem is to
evaporate an incoherent spin mixture of fermionic atoms as was used
to realize the first quantum degenerate Fermi gas with 40K (DeMarco
and Jin, 1999). Another strategy is by sympathetic cooling
(Wineland et al., 1978) a one-component Fermi gas that stay in
thermal contact with an actively cooled bath. This technique is
currently widely used, because the fermionic atom number is in
principle not reduced and high atom numbers with low temperatures
are achieved.
So far, successful sympathetic cooling of fermions has been
demonstrated with the follow- ing atomic combinations: 6Li − 7Li
(Truscott et al., 2001; Schreck et al., 2001), 6Li − 23Na
(Hadzibabic et al., 2002), 6Li− 87Rb (Silber et al., 2005), 40K−
87Rb (Roati et al., 2002; Kohl et al., 2005; Ospelkaus et al.,
2006b; Aubin et al., 2006) , 3He∗ − 4He∗ (McNamara et al., 2006)
and 171Yb − 174Yb (Fukuhara et al., 2007c).
The first experiments on degenerate Fermi gases were concentrated
on one-component systems, representing an ideal Fermi gas. These
studies explored, for example, thermodynamic properties of the
Fermi gas, its deviations from a classical gas, Pauli blocking of
collisions at ultralow temperatures (DeMarco and Jin, 1999; DeMarco
et al., 2001) and the Fermi pressure of a trapped gas (Truscott et
al., 2001).
In fermionic gases Feshbach resonances between two different
hyperfine states of an inco- herent mixture allow to tune the
interparticle interaction. Several Feshbach resonances were found
in 6Li (O’Hara et al., 2002b; Dieckmann et al., 2002; Jochim et
al., 2002) and 40K (Loftus et al., 2002). By a magnetic field sweep
across the Feshbach resonance weakly bound diatomic molecules were
produced (Regal et al., 2003). Surprisingly, these molecules showed
long lifetimes in the vicinity of a Feshbach resonance (Cubizolles
et al., 2003; Jochim et al., 2003a; Strecker et al., 2003; Regal et
al., 2004a), even for large interactions, which is contrary to the
case of bosons. The observed lifetimes were up to several seconds
at high densities of about 1013 cm−3.
This stability is a consequence of the Pauli exclusion principle
(Petrov et al., 2004), that suppresses vibrational quenching to
energetically lower vibrational molecular states. The increased
lifetime at resonance led to a boom of Fermi gases and allows to
work in the strongly interacting regime kF |a| ≥ 1, where the
scattering length a is larger or comparable to the interparticle
spacing ∼ 1/kF (kF is the Fermi momentum). This discovery opened
the way to explore the crossover regime between Bose-Einstein
condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS)
superfluidity (Bardeen et al., 1957), which is discussed in the
following Sec. 1.3.
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1.3 Many-body physics and the BEC-BCS crossover
To study many-body physics in dilute atomic gases the role of
interactions is an important is- sue. Periodic optical potentials
and Feshbach resonances, as previously mentioned in Sec. 1.1, are
essential tools to access the physics of strongly interacting
systems with atomic gases (see also review article on many-body
physics by Bloch et al. (2008)).
The ability to shape the dimensionality and to localize atoms with
optical potentials is one possibility to enter the regime of strong
correlations. An outstanding example is the quantum phase
transition from a superfluid to a Mott insulator with a BEC in a
three-dimensional optical lattice (Greiner et al., 2002). Other
examples with ultracold quantum gases in reduced dimensionality are
the realization of a Tonks-Girardeau gas (Kinoshita et al., 2004;
Paredes et al., 2004) in one-dimension and the
Berezinskii-Kosterlitz-Thouless crossover (Hadzibabic et al., 2006;
Schweikhard et al., 2007) in two-dimensions.
Moreover, bosons and fermions in periodic potentials (Modugno et
al., 2003) provide the link to condensed matter physics by the
(Bose-)Hubbard Hamiltonian. For example, the Fermi surface and the
band insulator with fermions are demonstrated (Kohl et al., 2005)
and a Mott insulator of fermions is recently realized (Jordens et
al., 2008; Schneider et al., 2008). The precise control of the
optical potentials offer novel directions in condensed matter
physics.
The other possibility to achieve strongly interacting systems are
Feshbach resonances that allow to control the interaction by tuning
the scattering length a. Long lifetime at the resonance in the bulk
gas is a necessary precondition for experimental studies, and only
is exhibited for the case of fermions, where decay is suppressed
due to Pauli exclusion principle (see discussion in Sec. 1.2). The
lifetime can even exceed the thermalization time by collisions.
This discovery allowed to create a molecular BEC (mBEC) (Jochim et
al., 2003b; Greiner et al., 2003; Zwierlein et al., 2003) of weakly
bound molecules composed of fermions on the repulsive, molecular
side of the Feshbach resonance (a > 0).
Furthermore, the long lifetime at large values of the scattering
length a is important to real- ize BCS pairing on the attractive
side of the Feshbach resonance (a < 0). In the weak coupling
regime kF |a| 1 the critical temperature TC scales with a as TC ≈
0.28TF exp (−π/2 kF |a|) (Gorkov and Melik-Barkhudarov, 1961).
Close to resonance, in the strong coupling regime kF |a| ≥ 1, no
analytic solution of the many-body problem is available and
numerical calcula- tions are required (Nozieres and Schmitt-Rink,
1985; Haussmann et al., 2007). In this regime TC is of order TF and
is thus experimentally accessible. This allowed to observe
condensation of fermionic pairs on the BCS-side (Regal et al.,
2004b; Zwierlein et al., 2004). Moreover, the tunability of the
scattering length allows to study the BEC-BCS crossover
(Bartenstein et al., 2004b; Bourdel et al., 2004), going from
pairing in real space to pairing in momentum space. This crossover
connects the two superfluid regimes smoothly across the strongly
interacting regime.
At resonance, the scattering length diverges and the gas is in the
so-called unitarity regime. Here, the only energy and length scale
of the system are the Fermi energy EF and Fermi momentum kF and all
thermodynamic quantities are related to universal properties
(Baker, 1999; Heiselberg, 2001; Ho, 2004).
Dynamic experiments at the BEC-BCS crossover include the
anisotropic expansion of the gas (O’Hara et al., 2002a) and
collective excitation measurements (Kinast et al., 2004;
Bartenstein et al., 2004a) with aiming to proof superfluidity.
However, similar behavior can be experimentally shown in a
classical hydrodynamic gas (Clancy et al., 2007; Wright et al.,
2007). A first signature of superfluid behavior along the crossover
was the observation
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1.4 Ultracold chemistry
of the pairing gap by radio-frequency spectroscopy (Chin et al.,
2004). The smoking gun experiment for superfluidity was the
observation of quantized vortices across the BEC-BCS crossover
(Zwierlein et al., 2005).
Currently, experiments concentrate on systems with imbalanced
spin-mixtures, that have an unequal occupation of the two spin
states. This leads to an unequal chemical potential of the two spin
components and exotic quantum phases with different pairing
mechanisms are expected in this situation (Fulde and Ferrell, 1964;
Larkin and Ovchinnikov, 1965; Sarma, 1963; Liu and Wilczek, 2003;
Bedaque et al., 2003; Caldas, 2004; Carlson and Reddy, 2005). In
imbalanced mixtures, so far, phase separation is observed
(Partridge et al., 2006), fermionic superfluidity with imbalanced
spin populations is proven (Zwierlein et al., 2006) and the
superfluid phase transition is directly observed (Schunck et al.,
2007).
1.4 Ultracold chemistry
Molecules at ultracold temperatures open new perspectives on high
precision tests of funda- mental physics, on quantum chemistry and
on novel quantum matter states (see also review article on
ultracold molecules by Carr et al. (2009)). Molecules provide
additional degrees of freedom by the rotational and vibrational
states, which offer new possibilities in their exper- imental
control. For the first time, ultracold molecules have the potential
to prepare a single quantum state of the internal and external
degrees of freedom and offer unique ways of high resolution quantum
control (Pe’er et al., 2007; Shapiro et al., 2008).
Moreover, dense and ultracold ensembles will allow precise control
of chemical reactions and the investigation of ultracold chemistry
(Krems, 2005; Hudson et al., 2006b; Krems, 2008). At high phase
space densities with large de Broglie wavelengths reaction dynamics
are influenced by quantum effects and tunneling processes play an
important role.
Of particular interest are ultracold polar molecules with the
dipole moment as an extra degree of freedom. In combination with
the precise controllability polar molecules can be used to test
fundamental physics, for example to study time variation of
fundamental constants (Hudson et al., 2006a; Chin et al., 2009) and
violation effects of parity and time-reversal (Ko- zlov and de
Mille, 2002; Hudson et al., 2002). Moreover, the dipole-dipole
interaction in polar molecular gases is anisotropic and has a
long-range character. This opens many new possibil- ities that
range from anisotropically interacting quantum fluids, like a polar
molecular BEC, to the exploration of electric dipole-dipole
mediated BCS pairing and many more phenomena (Santos et al., 2000,
2002; Goral and Santos, 2002; Goral et al., 2002; Baranov et al.,
2002; Damski et al., 2003; Griesmaier et al., 2006). Polar
molecules are also interesting for quan- tum information processing
(de Mille, 2002) due to effective coupling of the electric dipoles
at moderate distances and the relative easy controllability of the
electric dipole moments by electric fields (DC or microwave
fields).
The production process of ultracold molecular ensembles is an
experimentally challenging task and many different approaches
exist. The experimental techniques can be classified into direct or
indirect methods. Direct techniques start with stable ground-state
molecules that are further cooled in their external, motional
degree of freedom. Applying laser cooling methods to molecules
would be extremely demanding due to their complex level structure
and recent proposals, as (Bahns et al., 1996; Rosa, 2004; Stuhl et
al., 2008), are still to be tested. Current strategies of direct
cooling include decelerating a molecular beam by external fields,
which can be Stark (Bethlem et al., 1999; Bochinski et al., 2003)
or Zeeman (Hogan
5
1. Introduction
et al., 2007; Narevicius et al., 2008) potentials, or even optical
fields (Fulton et al., 2006). Another powerful and versatile
strategy is buffer-gas cooling with helium atoms (Weinstein et al.,
1998). The achievable temperatures with direct methods are
currently in the range of 10mK and 1K with relative small phase
space densities of up to 10−12. Therefore, to reach ultracold
temperatures, further cooling techniques have to be developed, e.g.
sympathetic cooling with laser cooled alkalines.
Indirect techniques, on the other hand, start with precooled atoms
by cooling methods previously described in Sec. 1.1 and 1.2 (i.e.
laser cooling, evaporative and sympathetic cool- ing). After
precooling, the atomic constituents are assembled to molecules by
photoassociative (Thorsheim et al., 1987) or Feshbach methods (Mies
et al., 2000). Due to technical constraints of trapping and laser
cooling, the indirect technique can be used for certain atomic
classes, such as alkali and alkaline earth metals. Typically, the
molecules that are created with this indirect method are in their
energetically highest vibrational state and can perform inelastic
relaxation to energetically deeper lying levels (Zirbel et al.,
2008a; Hudson et al., 2008), re- sulting in trap losses and
heating. Moreover, the expected electric dipole moments of these
heteronuclear molecules are negligible due to their highest
vibrational state. The excited molecules can be transferred to the
absolute rovibrational ground state (Sage et al., 2005), which is
energetically stable against inelastic two-body processes and
exhibits an appreciable electric dipole moment in the heteronuclear
case. (Note, that the expectation value of the electric dipole
moment is only non-zero when a finite electric field is
applied.)
The coldest temperatures are currently achieved by Feshbach methods
with temperatures down to a few tens of nanokelvin with high
phase-space densities and successful realization of BEC with
homonuclear molecules (Jochim et al., 2003b; Greiner et al., 2003;
Zwierlein et al., 2003). This method marks therefore the most
powerful and promising route towards quantum degenerate polar
molecules. At a Feshbach resonance weakly bound molecules can be
created by three different methods: The first one is by magnetic
field sweeps across a Feshbach resonance (van Abeelen and Verhar,
1999; Timmermans et al., 1999; Mies et al., 2000), the second one
by oscillatory fields (Thompson et al., 2005; Hanna et al., 2007)
and the third one by atom-molecule thermalization (Jochim et al.,
2003b; Zwierlein et al., 2003). So far, homonuclear bialkali
molecules have been produced of the bosonic species 23Na (Xu et
al., 2003), 87Rb (Durr et al., 2004) and 133Cs (Herbig et al.,
2003) and of the fermionic species 6Li (Strecker et al., 2003;
Cubizolles et al., 2003) and 40K (Regal et al., 2003). At present,
research is concentrated on heteronuclear bialkali molecules. The
molecules are currently produced from Bose-Fermi (40K − 87Rb
(Ospelkaus et al., 2006a; Zirbel et al., 2008b)) and from Bose-Bose
mixtures (85Rb − 87Rb (Papp and Wieman, 2006) and recently 41K −
87Rb (Weber et al., 2008)). These highly excitetd molecules can be
transferred to deeply bound states by a coherent stimulated Raman
adiabatic passage (STIRAP) (Bergmann et al., 1998), which allows
high transfer efficiencies. In 2008, homonuclear ground state
molecules of 87Rb2
(Lang et al., 2008) and 133Cs2 (Danzl et al., 2008) were created by
this coherent transfer technique. Furthermore, fermionic polar
molecules of 40K87Rb in their absolute ground state with phase
space densities close to quantum degeneracy were reported in the
same year (Ni et al., 2008).
6
1.5 This thesis
1.5 This thesis
This thesis describes two main achievements that offer new
perspectives for future experiments in the context of many-body
physics and ultracold chemistry.
Firstly, a quantum degenerate Fermi-Fermi mixture of two different
atomic species with unequal masses is realized for the first time.
This result adds the mass ratio and the different internal
structure as new control parameters to quantum degenerate Fermi
gases. Such a Fermi-Fermi mixture allows to study the nature and
existence of superfluidity in the presence of unmatched Fermi
surfaces, which can be realized by number imbalance or by mass
imbal- ance. In this situation, symmetric BCS pairing is broken and
many new quantum phases with different pairing mechanisms are
expected, as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state
(Fulde and Ferrell, 1964; Larkin and Ovchinnikov, 1965), the
breached pair state (Liu and Wilczek, 2003) and the Sarma state
(Sarma, 1963). Moreover, a crystalline phase transi- tion (Petrov
et al., 2007) and a link to baryonic phases of quantum
chromodynamics (Wilczek, 2007; Rapp et al., 2007) are
predicted.
Secondly, the first realization of ultracold heteronuclear bosonic
molecules of two different fermionic species is shown. This
completes the list of possible quantum-statistical combina- tions
in mixtures. The observed long lifetimes of the Fermi-Fermi
molecules in the molecule- atom mixture are promising to explore
many-body physics with two different masses and, in addition, to
create bosonic dipolar molecules in their absolute ground
state.
For experimental studies across a Feshbach resonance the molecular
lifetime is an impor- tant aspect. The stability of molecules
composed out of fermions depends on the mass ratio of the two
fermions (Petrov et al., 2005). At a Feshbach resonance long
molecular lifetimes are expected for mass ratios smaller than
12.33, for which the molecular relaxation decreases with an
increasing scattering length. For the experimental studies
therefore, the fermionic isotopes of potassium and lithium are
chosen (40K and 6Li) that have a mass ratio of 6.67.
To cool the Fermi-Fermi mixture into quantum degeneracy we chose
the sympathetic cooling strategy, which uses another atomic cloud
as a cooling agent. In our case we decided to use the bosonic
alkali atom 87Rb. Sympathetic cooling has the advantage that the
atom number of the fermions is in principle not reduced by
evaporation, and therefore avoids the need for high number atom
sources. Moreover, evaporation of fermions would have the
disadvantage of Pauli blocking at the final stage of evaporation.
In addition, the expected dipole moment for ground state molecules
of LiK and LiRb (Aymar and Dulieu, 2005) is larger than in
comparable existing experiments and should allow to study quantum
gases with large dipolar interactions.
The following chapters of this thesis are organized as follows:
Chapter 2 gives an theoretical overview on the quantum statistical
and thermodynamic properties of bosons and fermions. Then
collisional processes in ultracold gases and Feshbach resonances
are discussed. After that, Chapter 3 describes the experimental
platform to explore a quantum degen- erate Fermi-Fermi-Bose mixture
and to investigate ultracold heteronuclear Fermi-Fermi molecules.
The discussion begins with the experimental concept, followed by
the vac- uum system and the atomic sources. After that I give an
overview of the laser system for trapping and manipulating the
atoms. Then the imaging system and the setup for magnetic trapping
are described, which is followed by the presentation of an ultra
stable
7
1. Introduction
current control for the magnetic Feshbach fields. After that I show
the setup for radio- frequency and microwave manipulation and
finally I explain the experimental control in real-time. In Chapter
4 I present the first magneto-optical trapping of two different
fermionic species, 6Li and 40K, and a bosonic species 87Rb. This
also demonstrates simultaneous trapping of three species in a
magneto-optical trap (“triple MOT”) for the first time.
Optimization of the atom numbers in the triple MOT configuration is
shown and, in addition, loading and trap losses are characterized.
These results pave the way towards a quantum degenerate mixture of
two different fermionic species. Parts of this Chapter are
published in (Taglieber et al., 2006). The first quantum degenerate
mixture of two fermionic species, 6Li and 40K, is presented in
Chapter 5. I will give an overview on the experimental way to
achieve such a quan- tum degenerate Fermi-Fermi mixture and show
crucial experimental challenges. The two fermions are
sympathetically cooled by a bosonic 87Rb gas. In addition, a
quantum degenerate two-species Fermi-Fermi mixture coexisting with
a BEC is realized, repre- senting simultaneous degeneracy of three
species for the first time. Furthermore, an increased cooling
efficiency for 6Li by 87Rb in the presence of 40K is observed,
demon- strating catalytic cooling. Parts of this Chapter are
published in (Taglieber et al., 2008). Finally, Chapter 6 presents
the first creation of ultracold bosonic heteronuclear molecules of
two fermionic species, 6Li and 40K, by a magnetic field sweep
across an interspecies s-wave Feshbach resonance. First, a crossed
optical dipole trap is characterized and the loading sequence of
the atoms into this trap is described. After that, magnetic field
calibration of the Feshbach coils and state preparation of 6Li and
40K to the desired states are explained. The discussion is followed
by identifying Feshbach resonances and ends with measurements on
heteronuclear Fermi-Fermi molecules. Parts of this chapter are
published in (Voigt et al., 2009).
8
2.1 Ultracold gases
This theoretical part briefly summarizes the quantum statistical
and thermodynamic prop- erties of bosons and fermions. In the
following, the main formulas for Bose-Einstein con- densation and
degenerate fermions are presented that allow to extract the
essential physical parameters in an absorption measurement.
A detailed description can be found in several textbooks, e.g.
(Cohen-Tannoudji et al., 1977a,b; Sakurai, 1994; Huang, 1987;
Pitaevskii and Stringari, 2003).
2.1.1 Quantum statistical properties
In this chapter the basic quantum statistical and thermodynamic
properties of bosons and fermions are discussed.
The total quantum mechanical wavefunction of identical particles is
either symmetric or anti-symmetric under exchange of two particles.
This leads to the classification of bosonic and fermionic particles
for the symmetric or anti-symmetric case, respectively. Given the
thermodynamic partition function for non-interacting particles, the
mean occupation number f (εr) of a single particle energy
eigenstate with energy εr can be deduced and is given by
f(εr) = 1
. (2.1)
Here, β = (kBT )−1 specifies the temperature T of the system and µ
is the chemical po- tential. The Fermi-Dirac statistics describes
fermions, the Bose-Einstein statistics bosons and the
Maxwell-Boltzmann statistics a classical system. For a given atom
number N , the normalization condition of the sum over all possible
states
N = ∑
r
f(εr) (2.2)
fixes the chemical potential µ. Fig. 2.1 shows the chemical
potential for bosons, fermions and distinguishable classical
particles as a function of the temperature.
In experiments, the atomic ensemble is typically held in a harmonic
potential Vho (r). This case is discussed and assumed in the
following. A particle of mass m experiences the potential
9
Figure 2.1: The chemical potential for bosons, fermions and
distinguishable classical particles vs. temperature in a harmonic
confinement. The critical temperature Tc
for BEC is related to TF by Tc/TF = (6 g3(1)) −1/3 ≈ 0.52.
Vho (r) = m
2 )
, (2.3)
where ωi (i = x, y, z) are the angular trapping frequencies. The
mean trapping frequency is
then given by ω = (ωx ωy ωz) 1/3. For large atom number and when
the level spacing greatly
exceeds the thermal energy kBT ~ωi (i = x, y, z), the discrete sum
in Eq. (2.2) can be replaced by an integral over the product
between the mean occupation number f(ε) and the density of states
g(ε). For harmonic trapping potentials the density of states is
given by
g(ε) = ε2
2 (~ ω)3 , (2.4)
which is exact in the thermodynamic limit. This expression fails
for bosons at low tempera- tures, where the ground state is
macroscopically occupied, but is weighted to zero. Therefore, the
ground state for bosons has to be taken separately into account. On
the other hand, identical fermions can only occupy at most one
state and the ground state can be neglected for large atom numbers.
The number of atoms in excited states is then given by
Nex =
0 f(ε) g(ε) dε . (2.5)
The integration can be calculated analytically and gives for the
bosonic and fermionic cases
Nex = −θ
g3(−θ z) . (2.6)
Here, z = eβ µ defines the fugacity parameter. gα(s) is the
polylogarithm function and is given by the relation
gα(s) =
kα . (2.7)
It is defined for α, s ∈ C , |s| < 1 and its analytic
continuation.
10
2.1.2.1 Bose-Einstein condensate
The number of bosonic atoms for given temperature and trapping
parameters in the excited states is a finite value. The system
accommodates more particles in excited states as µ approaches zero.
At this point additional atoms macroscopically occupy the ground
state, while the chemical potential stays locked to zero. This
phenomenon is called Bose-Einstein condensation (BEC). If N is the
total atom number, then the atom number of the BEC is given
by
N0 = N −Nex . (2.8)
For non-interacting particles the critical temperature Tc for a BEC
can be deduced from Eq. (2.6) by setting N0 = 0 and µ = 0 :
Tc = ~ ω
kB N1/3 . (2.9)
The condensate fraction is a function of temperature and for T <
Tc the fraction, which can be deduced from Eq. (2.6) and (2.9), is
given by
N0
T
Tc
)3
. (2.10)
The density distribution of the trapped gas is calculated in Sec.
2.1.2.3 below. From Eq. (2.20) follows that BEC occurs if the peak
density of the distribution fulfills the following condition:
n0 λ 3 dB = g3/2(1) ≈ 2.612 , (2.11)
where
λdB =
2π ~2
mkBT (2.12)
is the thermal de Broglie wavelength. This means that BEC occurs
when λdB is on the order ofthe mean particle separation and the
indistinguishability of the particles is thus shown.
2.1.2.2 Weaky interacting Bose-Einstein condensate
Ultracold and dilute gases interact via elastic interaction and
only binary interactions are relevant. At low temperatures
collisions between particles occur in the s-wave scattering limit.
The interaction can be written by a zero-range pseudo-potential V
(r− r′) = g δ (r− r′) (Huang, 1987), where
g = 4π ~
2 a
m (2.13)
describes the coupling constant. In this equation a is the s-wave
scattering length (see also discussion in Sec. 2.2.1). For 87Rb the
interaction is repulsive and the triplet scattering length is
106(4) a0.
In a mean field description the condensate wavefunction can be
described by a macro- scopic wavefunction Ψ(r, t), which has the
meaning of an order parameter. The density of
11
2. Theory
the condensate is then given by nc(r, t) = |Ψ(r, t)|2. In a
variational approach, where a min- imization of the energy is
performed, the Gross-Pitaevskii equation (GPE) can be derived. The
time-independent GPE (Goldman and Silvera, 1981; Huse and Siggia,
1982) is given by
µΨ(r) =
)
Ψ(r) . (2.14)
The equation is valid for temperatures T Tc, large atom number N0 1
and weakly interacting particles with nc |a|
3 1. If g nc (r) ~ωi (i = x, y, z) for repulsive interactions (i.e
a > 0), one can neglect the kinetic term of Eq. (2.14). This
limit called the Thomas-Fermi approximation. The condensate density
distribution is then given by
nc(r) = max
. (2.15)
For harmonic potentials the density distribution has a parabolic
shape and vanishes when the chemical potential is equal to the
trapping potential. The Thomas-Fermi radius of the condensate in
the i direction is
Ri =
. (2.16)
The atom number of the condensate can be calculated by integrating
over nc(r) and is given by
N0 =
~/m ω is the oscillator length.
2.1.2.3 Density distribution and free expansion
Non-condensed gas The density distribution of the non-condensed
part of the bosonic gas can be obtained by a semiclassical
approximation of the phase space density distribution ρ(r, p).
Here, a wave packet with definite position and momentum is assigned
to each particle. The phase space density distribution is given
by
ρ(r, p) = 1
eβ(p2/2 m+V(r)−µ) − 1 . (2.18)
In the experiment atomic clouds are typically probed after they are
released from the trapping potential. For an arbitrary free
expansion time the density of the thermal component can easily be
deduced. For a harmonic trap it is given by
n(r, t) =
. (2.20)
2.1 Ultracold gases
Figure 2.2: The Fermi-Dirac distribution function vs. energy for
three different temperatures. In the T → 0 limit, all energy states
are occupied up to the Fermi energy EF = µ(T → 0, N).
Bose-Einstein condensate The expansion of a Bose-Einstein
condensate in a radial sym- metric, cigar-shaped trap evolves
differently in comparison to the thermal part. The temporal
evolution can be calculated from the GPE (Castin and Dum, 1996;
Kagan et al., 1996; Dalfovo et al., 1997). The trapping frequencies
are assumed to be ωρ along the radial direction and ωz
along the axial direction. Introducing the dimensionless parameters
τ = ωρ t and λ = ωρ/ωz, the Thomas-Fermi radii in radial and axial
direction evolve as
Rρ(t) = Rρ(0) √
. (2.22)
These two equations show an anisotropic expansion from a
cigar-shaped to a pancake-shaped cloud.
2.1.3 Fermions
2.1.3.1 Fermi energy
Spin-polarized fermions can only occupy one state, which is a
consequence of the Pauli ex- clusion principle. When a Fermi gas is
cooled towards lower temperatures, the Fermi-Dirac distribution
function fFD becomes relevant. For T → 0 all energy states are
filled up to the Fermi energy EF = µ(T → 0, N), which is defined by
the chemical potential. The correspond- ing Fermi temperature is
given by TF = EF/kB. Fig. 2.2 shows the Fermi-Dirac distribution
function for different temperatures. In the T → 0 limit, fFD
evolves into a Heaviside step function that is 1 for temperatures
below TF and 0 above it.
The Fermi energy can be derived for a harmonic trap by Eq. (2.5)
and gives
EF = ~ ω (6N)1/3 . (2.23)
In an experimental sequence the temperature of a Fermi gas is
typically extracted in free
13
2. Theory
expansion. Here, the relation between the temperature and the
fugacity is useful:
T
TF =
2.1.3.2 Density distribution and free expansion
The density distribution of a fermionic cloud can be derived from
the phase space density distribution, as in the bosonic case. For
details see also (Pitaevskii and Stringari, 2003). A Fermi gas at T
= 0 has the following density distribution:
n(r, T = 0) = (2m)3/2
6π2 ~3 (EF − V (r))3/2
and is zero for EF < V (r). For a harmonic trap the cloud size
in the i direction can be described by the Fermi radius
Ri =
. (2.25)
In a free expansion measurement the density distribution for an
arbitrary time-of-flight and temperature is given by
n(r, t) = −
. (2.26)
This density distribution for fermions and the one for bosons,
which is given by Eq. (2.20), differ only by the minus signs.
2.2 Feshbach resonances
The second theoretical part gives describes the physics of
ultracold collisions in dilute gases and briefly discusses Feshbach
resonances.
2.2.1 Ultracold Collisions
This section presents the basic principles of elastic collisions in
ultracold, dilute gases. It shows that in the zero energy limit the
scattering process can be described by one single parameter, the
s-wave scattering length.
For details I refer to the following literature (Taylor, 1972;
Landau and Lifschitz, 1985; Sakurai, 1994; Dalibard, 1999;
Pitaevskii and Stringari, 2003).
2.2.1.1 Elastic collisions
In ultracold, dilute systems the thermal de Broglie wavelength λdB
and the interparticle distance n−1/3 are typically much larger than
the range of the interatomic potential r0, which is on the order of
the van der Waals length lvdW. Therefore, for the description of
the scattering process at a potential V (r), the short-range
scattering potential can be neglected.
14
2.2 Feshbach resonances
(
− ~
Ψk(r) = EΨk(r) . (2.27)
Here, mr = m1m2/ (m1 +m2) is the reduced mass and E = ~ 2 k2/ (2mr)
is the energy of the
plane wave. For large distances, r → ∞, the wavefunction Ψk(r)
possesses the asymptotic form
Ψk(r) ∼ eik·r + f(k, ϑ, ) eikr
r , (2.28)
with the spherical coordinates r, ϑ, . This expression corresponds
to a sum of an incom- ing plane wave and an outgoing scattered part
that is described by a scattering amplitude f(k, ϑ, ). The cross
section is given by
σ(k) =
|f(k, ϑ, )|2 d . (2.29)
For a central potential one can perform the standard expansion of
the incident and out- going wave into partial waves with angular
momenta l. Here, l stands for s-, p-, d-, ... waves. Scattering at
a central potential does not change l, but induces a phase shift
δl. This allows to express the cross section σ(k) as a sum over all
partial wave cross sections (Landau and Lifschitz, 1985)
σ(k) =
2.2.1.2 Low-energy scattering (Zero-energy limit)
In ultracold gases with temperatures of up to several 100µK the
scattering process is described by low momenta k 1/r0. The
scattering phase modulo 2π scales as δl ∝ k2l+1 for k → 0 (Landau
and Lifschitz, 1985). Thus, the partial wave cross section with
angular momenta l has the scaling
σl ∝ k4l ∝ E2l (2.31)
for k → 0. This shows, that in the low temperature limit, the
s-wave collisions are dominant and collisions of angular momentum l
≥ 1 are strongly suppressed. For k 1/r0 the effective- range
expansion to second order in k gives (Landau and Lifschitz,
1985)
k cot δ0 ≈ − 1
tan δ0(k)
k , (2.33)
which defines the effective range reff of the scattering potential
and the scattering length a. The effective range reff for atom-atom
interactions with large interatomic separations, that can be
described by the van der Waals potential V (r) = −C6/r
6, is on the order of the van der Waals length lvdW (Flambaum et
al., 1999) that is given by
lvdW = 1
d
Figure 2.3: The graph shows two molecular potentials Vbg (R) and Vc
(R) vs. the interatomic distance. Eδ is the threshold energy
difference between a molecular bound state and the open channel. If
the open and closed channel couple to each other, the bound
molecular state can be occupied and the scattering is resonantly
enhanced.
In general, the scattering length a is not restricted and can be −∞
< a < +∞. With the above expansion the scattering amplitude
can be expressed as
f(k) = 1
. (2.35)
In the case of k |a| 1 and reff 1/k the scattering amplitude
depends only on momen- tum and yields f = i/k. This regime is
called the unitarity limit and the cross section of distinguishable
particles is σ = 4π/k2.
For k |a| 1 and |reff| . 1/k the scattering amplitude is a function
of one single pa- rameter, f = −a, and is independent on momentum.
In this zero energy limit the scattering cross section of
distinguishable particles is σ = 4π a2. For identical particles the
scattering amplitudes interfere and the total wavefunction has to
be symmetrized or anti-symmetrized. This leads to doubling of the
symmetric even partial waves contributions for the bosons and
cancelling of the anti-symmetric odd partial waves. For fermions,
on the other hand, the odd partial waves contributions are doubled
and the even ones are cancelled. Therefore, identical fermions do
not scatter in the zero energy limit and form an ideal gas.
Summarized, the s-wave cross section is given by
σ =
2.2.2 Magnetic Feshbach resonance
The collision process, so far discussed, contains only a single
interaction potential. In general, binary collisions in ultracold
gases are more complex due to different internal states of the
particles. One possible consequence is that the collision partners
might get resonant to a molecular bound state and form a Feshbach
resonance (Fano, 1935, 1961; Feshbach, 1958,
16
2.2 Feshbach resonances
Figure 2.4: (a) In the vicinity of a Feshbach resonance the
coupled, bound molecular state and the uncoupled, bare molecular
state is plotted vs. the magnetic field. Due to interchannel
coupling, the resonance position of the dressed molecular state
B0
is shifted with respect to the threshold crossing Bres of the bare,
uncoupled state. The shaded area above the dissociation threshold
represents the continuum. (b) The scattering length is plotted as a
function of the magnetic field. At the resonance position the
scattering length has a pole, a→ ±∞.
1962). In ultracold quantum gases the Feshbach resonance is a key
tool for many breakthrough experiments. With a Feshbach resonance
the interactions of quantum matter can be controlled precisely and
the study many-body physics and condensed matter phenomena are
possible. Moreover, molecular physics in the zero temperature limit
is accessible.
In the following the basic principle of a magnetically induced
Feshbach resonance is dis- cussed. For details I refer to the
review articles (Kohler et al., 2006; Chin et al., in prepara-
tion).
2.2.2.1 Principle
A simple model of the Feshbach resonance takes two molecular
potentials Vbg(R) and Vc(R) into account. Both of them have a
different threshold energy. For small kinetic energies E two free
atoms with large interatomic distance approach each other in the
so-called entrance channel or open channel and enter the background
potential Vbg(R). Due to the small kinetic energy the upper
potential Vc(R) is energetically closed and is called the closed
channel.
This closed channel can have bound molecular states that are near
the threshold or dis- sociation energy of the open channel. If the
open and closed channel couple to each other, the bound molecular
state can be occupied and the scattering is resonantly enhanced.
The molecular potentials are illustrated in Fig. 2.3.
In general the magnetic moments of the bare bound molecular state
µmol and of the asymptotically separated atoms µatoms are
different. µres is defined as the difference between these two
magnetic moments. Then the energy difference of the two states,
given to first order by
Eδ = µres (B −Bres) , (2.37)
can easily be varied by an external magnetic field of strength B.
In the above expression the
17
2. Theory
magnetic field Bres is the threshold crossing of the bare,
uncoupled state. This tunability leads to magnetically induced
Feshbach resonances, which allows a precise control of the
scattering length. For s-wave collisions the scattering length has
a simple form and is given by (Moerdijk et al., 1995)
a(B) = abg
. (2.38)
This expression only depends on three parameters: abg is the
background scattering length of the potential Vbg(R), is the
resonance width and B0 is the resonance position of the dressed
molecular state, where a(B) diverges. B0 is shifted with respect to
Bres due to interchannel coupling and the shift is given by the
relation (Kohler et al., 2006)
B0 −Bres = abg
1 + (1 − abg/ a) 2 . (2.39)
Here, a defines a mean scattering length that is defined as
a = 4π
Γ(1/4)2 lvdW ≈ 0.95598 lvdW . (2.40)
Fig. 2.4 shows the molecular states and the scattering length
dependency on the magnetic field.
2.2.3 The asymptotic bound state model
A realistic model for the interaction of two alkali atoms includes
the relative electronic spin orientation. In this section, the
asymptotic bound state model (Wille et al., 2008) is discussed,
which is based on the models by (Moerdijk et al., 1995) and (Stan
et al., 2004) and is extended by a mixing term between singlet and
triplet states (see also (Walraven, 2009)). The coupled- channel
calculation (Stoof et al., 1988), which is computationally more
demanding, takes the exact interaction potentials and the coupling
between spin channels into account.
In the Born-Oppenheimer approximation the electronic motion is
effectively decoupled from the nuclear motion. At ultracold
temperatures the molecule is in its electronic ground state
potential and has a zero angular momentum. The electronic spins of
the two colliding alkali atoms are in a S-state and are either in
the singlet or triplet state. The relevant singlet potential
Vs=0(r) is then X1Σ+ and the triplet potential Vs=1(r) is a3Σ+.
These potentials are isotropic, which allows to write the relative
motion of the atoms through the following Hamiltonian:
Hrel = ~
2
2mr
S=0,1
Vs(r)Ps . (2.41)
Here l is the projection quantum number of the angular momentum, mr
is the reduced mass, r is the interatomic separation and Ps is the
projection operator onto the singlet or triplet spin states.
The total Hamiltonian that includes the hyperfine structure is
given by
H = Hhf +HZ +Hrel . (2.42)
The two first terms represent the hyperfine and Zeeman energies of
each atom and are given by
18
2 i2 ·B , (2.44)
where ahf Li and ahf
K are the hyperfine coupling constants of 6Li and 40K. s is the
single-atom electron spin and i is the corresponding nuclear spin.
The total electronic spin of the two atoms is S = s1 + s2. γ
e and γn are the gyromagnetic ratios of the electron and nuclei.
The hyperfine Hamiltonian Hhf can be rewritten in terms of (sLi +
sK) and (sLi − sK) and
gives Hhf = Hhf
Hhf - =
ahf 2
2 ~2 (s1 − s2) · i2 . (2.47)
The Hamiltonian Hhf + contains terms proportional to the total
electronic spin S, and thus
preserves the separation of the orbital from the spin problem. Hhf
- , however, mixes singlet
and triplet states. The full Hamiltonian of Eq. (2.42) conserves
the quantum numbers l and the total angular
ψl S
describes the asymptotic last bound state of l (l + 1) / (
2mr r 2 )
+ VS . The second part |S, MS , m1, m2 describes the spin
function.
⟨
⟩
⟨
= 0 . (2.48)
Here El S is the energy of the last bound state with angular
momentum l and spin S. The
energy El=0 S corresponds to the singlet and triplet scattering
lengths. It can be shown, that
including additional information about the potential shape, the
bound state energies El S with
l > 0 can be deduced from El=0 S (Wille et al., 2008).
2.2.3.1 Feshbach resonances in the 6Li-40K mixture
The 6Li-40K mixture offers several s- and p-wave Feshbach
resonances. The last bound state energies El=0
S can be derived by fitting the observed resonances with the
solutions of Eq. (2.48). According to (Wille et al., 2008), the
parameters are El=0
S=0 = h× 716 (15) MHz and El=0 S=1 =
h× 425 (5) MHz. As an example the bound state energies for the MF =
−2 collision channel between
6Li |F = 1/2, mF = 1/2 and 40K |F = 9/2, mF = −5/2 are calculated.
Fig. 2.5 shows the molecular bound state energies for s- and
p-waves and the open channel.
19
2. Theory
Figure 2.5: The bound state energies for the MF = −2 collision
channel between 6Li |F = 1/2, mF = 1/2 and 40K |F = 9/2, mF = −5/2.
The bound state energies for s- and p-waves (l = 0 and l = 1) and
the open channel are plotted versus the magnetic field. The
crossings between the molecular bound states and the open channel
give the position of the Feshbach resonances.
The crossings between the molecular bound states and the open
channel give the position of the Feshbach resonances. In the MF =
−2 case three s-wave and one p-wave Feshbach resonances are
experimentally observed. The asymptotic bound state model predicts
one ad- ditional, not observed p-wave resonance at 17.5G. The group
of Prof. E. Tiemann calculated the resonances with a more accurate
model recently and reported, that this resonance does not appear
(Tiemann et al., 2009).
2.2.4 Classification of Feshbach resonances
The physics at a Feshbach resonance is in principle affected by the
open and closed channel states. However, it can be shown that for
certain parameters the closed channel contributions can be
neglected and, thus, the physical description of the system is
independent of the properties of the molecular state. In this
sense, the system is suitable to realize universal many-body
phenomena. In the following, the conditions of universal behavior
are discussed for fermionic systems (for details see (Sheehy and
Radzihovsky, 2007; Ketterle and Zwierlein, 2008)).
The relevant energy scales are the Fermi energy EF, the energy
difference Eδ of the open and closed channel and finally the
coupling energy E0 of the Feshbach resonance. E0 can be expressed
by experimental measurable observables (Ketterle and Zwierlein,
2008):
E0 = 1
. (2.49)
On the molecular or BEC side of the Feshbach resonance (a > 0)
the dressed molecular state is a superposition of the closed and
open channel components. The fraction Z(B) of the closed channel
part can be related to the difference in magnetic moments of the
two channels
20
Z(B) = 1
∂B . (2.50)
Here Eb is the binding energy of the bound molecular state. Close
to resonance, in the universal regime, the binding energy of the
molecule depends solely on the scattering length a (Kohler et al.,
2006):
Eb = − ~
2
ma2 . (2.51)
In this regime the molecule forms a “halo” dimer with a bond length
of r = a/2 that greatly exceeds the outer classical turning point
for a lvdW. Combining Eq. (2.50) and (2.51) gives together with Eq.
(2.49) and the Fermi energy EF the following condition:
Z(B) = 2
kF a . (2.52)
. (2.53)
On the atomic or BCS side of the Feshbach resonance (a < 0) the
system is still affected by the closed channel even though the
bound molecular state disappears. A finite-energy resonance appears
for Eδ > E0 in the scattering cross-section (Andreev et al.,
2004; Gurarie and Radzihovsky, 2007). This resonant state has a
peak energy at
E∗ = Eδ − E0 (2.54)
with
Thus, scattering at the finite-energy resonance is
non-universal.
On resonance the effective range of the scattering potential has
the value (Ketterle and Zwierlein, 2008)
reff = −
2 ~2
mE0 (2.56)
and effects the scattering amplitude of Eq. (2.35) by the 1 2 reff
k
2 contribution.
For universal behavior the physics should not depend on the
molecular state for energies |Eδ| ≤ EF. Furthermore, molecular
effects throughout the entire strongly interacting regime kF |a| ≥
1 should be negligible. Applying these two constraints on the above
three regimes, one gets the universality condition E0 EF or
equivalently kF reff 1. This means that the interparticle
separation has to be much larger than the effective range of the
scattering po- tential. Such a Feshbach resonance is called broad
resonance, because E0 scales quadratically with the resonance width
and the condition is usually fulfilled for resonances much larger
than 1G.
21
2. Theory
Figure 2.6: The molecular relaxation rate αrel is strongly affected
by quantum statistics. In fermionic systems the relaxation rate
scales as αrel ∝ a−s. The figure shows the exponent s for
dimer-dimer collisions as a function of the mass ratio M/m of the
fermionic atoms.
In the other case, E0 EF, the resonance is called narrow and the
molecular state can not be neglected and has to be included in the
many-body description (De Palo et al., 2004; Simonucci et al.,
2005; Sheehy and Radzihovsky, 2007). At such a narrow resonance
molecules in the closed channel state can already form on the BCS
side (a < 0) for Eδ > 0 (Falco and Stoof, 2004; Gurarie and
Radzihovsky, 2007). Due to equilibrium of the fermions and
molecules the chemical potential holds the condition 2µ ≤ Eδ.
Hence, it is energetically favorable to convert a certain fraction
of the fermions into molecules in the regime 0 < Eδ < 2EF. It
is expected that the decay of these molecules into free atoms is
Pauli blocked due to the presence of the Fermi sea (Falco and
Stoof, 2004).
2.2.5 Fermi-Fermi mixtures
The collisional properties of Feshbach molecules crucially depend
on the quantum statistical properties of the corresponding
constituents. For Fermi-Fermi molecules a surprisingly long
lifetime at high densities of about 1013 cm−3 was observed in the
vicinity of a Feshbach resonance (Cubizolles et al., 2003; Jochim
et al., 2003a; Strecker et al., 2003; Regal et al., 2004a). The
molecular lifetime was about 100ms for 40K and several seconds for
6Li. The mechanism, explained by (Petrov et al., 2004) for halo
dimers, is briefly discussed in the following.
The increased lifetime is attributed to a suppression of inelastic
decay by vibrational quenching to lower vibrational molecular
states. The corresponding mechanism is related to two facts: First,
the wavefunction overlap between a Feshbach molecule with a large
spatial extension and more deeply bound molecular states is small;
second, Pauli suppression holds for the collisional process between
fermions. A halo dimer has a characteristic size that is given by
the scattering length a. The spatial size of deeply bound molecular
states is r0, which is on the order of the van der Waals length
lvdW and is much smaller than a close to resonance. A relaxation to
deeply bound states requires at least three atoms that come close
at a distance r0. Since two of the three fermionic atoms are
necessarily identical, the relative wavefunction
22
2.2 Feshbach resonances
has to be antisymmetric. Consequently, the wavefunction has a node
at the relative distance r = 0 and varies for small values of r as
∼ k r. The characteristic momentum spread of the atoms of the halo
dimer is k ∼ 1/a. Therefore, the relaxation probability is
suppressed by a certain power of (k r0) ∼ (r0/a).
The exact relaxation rate αrel has been first calculated for equal
masses (Petrov et al., 2004) and has been extended for unequal ones
(Petrov et al., 2005). The derivation assumes a negligible
component of the closed channel part and a binding energy that
obeys the universal scaling with the scattering length a according
to Eq. (2.51). αrel is found to be
αrel = C ~ r0 m
(r0 a
)s . (2.57)
Here, the coefficient C is a system dependent parameter and s is an
exponent that depends on the collisional partners. This exponent
for dimer-dimer collisions is calculated according to (Petrov et
al., 2005) and is plotted in Fig. 2.6. For equal masses M = m of
the fermionic atoms the exponent is s = 2.55 and continuously
decreases for larger mass ratios. For 40K and 6Li s = 1.39 and is
zero at a mass ratio of M/m = 12.33. For larger mass ratios the
collisional relaxation increases up to M/m = 13.6. For larger mass
ratios M/m > 13.6 a short-range three-body parameter is
necessary for the calculation. However, for very large mass ratios
M/m > 100 a gas-crystal quantum transition is predicted (Petrov
et al., 2007).
In contrast to fermions, bosons show an increased relaxation rate
at a Feshbach resonance that scales as a4 with the scattering
length (Fedichev et al., 1996b; Weber et al., 2003; Petrov, 2004),
even though the wavefunction overlap between the initial halo dimer
state and final deeply bound molecular states decreases.
23
Chapter 3
Experimental Setup
This chapter describes the experimental platform to explore a
quantum degenerate Fermi- Fermi-Bose mixture and to investigate
ultracold heteronuclear Fermi-Fermi molecules. The design and
realization of this platform is one of the main parts of this
thesis.
In the following sections I will describe the experimental
apparatus in detail. The dis- cussion begins with the experimental
concept in Sec. 3.1, followed by the vacuum system in Sec. 3.2 and
the atomic sources in Sec. 3.3. After that I give an overview of
the laser system for trapping and manipulating the atoms, Sec. 3.4.
The imaging system is described in Sec. 3.5, followed by the setup
for magnetic trapping, Sec. 3.6. An ultra stable current control
for the magnetic Feshbach fields is explained in Sec. 3.7. Then I
show the setup for radio-frequency and microwave manipulation in
Sec. 3.9 and finally I explain the experimental control in
real-time in Sec. 3.10.
3.1 Experimental concept
The main idea of the experimental platform is to explore quantum
degenerate mixtures of two atomic fermionic species. Such a
Fermi-Fermi system is of particular interest in the vicinity of a
Feshbach resonance. Here it is possible to create heteronuclear
bosonic molecules, which can either be used to create dipolar
molecules or to explore the BEC-BCS crossover.
In our experiment we decided to use the fermionic alkali atoms 6Li
and 40K. For cooling into the quantum degenerate regime we chose
the strategy by sympathetic cooling the two fermions by a cooling
agent, namely the bosonic alkali atom 87Rb. This avoids the
reduction in atom number of the fermions by evaporation, and
therefore the need for high number atom sources. Moreover,
evaporation of fermions would have the disadvantage of Pauli
blocking at the final stage of evaporation.
From the experimental point of view, the selected species have
special advantages. All three species are well-known and have a
simple energy structure. They have been cooled to quantum
degeneracy in single species configuration already, but not
together. After designing, building-up the system and working on
quantum degeneracy, even the combination of 6Li and 87Rb have not
been realized before. All species are alkali atoms and therefore
one expects several interspecies Feshbach resonances. The necessary
laser systems for trapping, cooling and manipulation can be built
up by cost-effective semiconductor laser sources. The laser
wavelengths are close together and allow to use common
optics.
The concept of the experimental setup to achieve quantum degeneracy
is as follows: First,
25
oven chamber
MOT chamber
UHV chamber
Figure 3.1: The vacuum system for experimental studies with the
atomic species lithium, potassium and rubidium consists of three
different parts: the oven chamber, the MOT chamber and the UHV
chamber.
the atoms are loaded simultaneously into a magneto-optical trap
(MOT) in a vacuum chamber (“MOT chamber”), lithium from a Zeeman
slower, potassium and rubidium from background vapor pressure. To
achieve longer lifetimes, necessary for evaporative and sympathetic
cool- ing, the atoms are transferred into an ultra high vacuum
chamber (“UHV chamber”). For excellent optical access, the transfer
is realized by a magnetic transport (Greiner et al., 2001) around a
corner into a glass cell. Here the atoms are transferred into a
Ioffe-type trap to suppress Majorana losses. Through selective
evaporative cooling of rubidium, the fermions are sympathetically
cooled to quantum degeneracy. For studies with Feshbach resonances,
the atoms are loaded into an optical dipole trap.
3.2 Vacuum system
For experiments with atomic quantum gases in magnetic traps based
on coils ultra-high vac- uum, typically below 1 × 10−11 mbar, is
needed. This reduces losses and heating from back- ground gas
during the experimental cycle. However, for MOT loading from vapor
dispensers high partial pressure is required. Therefore, the vacuum
system consists of three different parts, the oven chamber for
lithium, the MOT chamber and the UHV chamber (see Fig. 3.1).
The oven chamber is used for loading lithium with a Zeeman slower
(Phillips and Metcalf, 1982) and gives a collimated atomic beam
(for details to the Zeeman slower see the following Sec. 3.3). This
chamber starts with a lithium oven that can be heated up to several
100 °C in a controlled way (for details see Voigt (2004)). The oven
is connected to a five-way connector through a differential pumping
stage (6mm inner diameter, 23 cm length) to reduce the atomic
26
3.2 Vacuum system
flux into this chamber. The atomic beam can be blocked by an atomic
beam shutter, connected to a rotation feed-through. For optical
analysis of the atomic beam two windows are flanched to the
five-way connector. The oven chamber is pumped by an 50 l/s ion
pump (Varian, VacIon Plus 55 StarCell) and connects the MOT chamber
through a second differential pumping stage, two tubes with an
inner diameter of 6mm and a length of 16.5 cm. Between the tubes, a
pneumatically actuated valve interrupts the vacuum connection and
allows refilling the lithium oven without breaking the whole vacuum
system.
A steel tube for the Zeeman slower (77 cm long) connects the valve
with the MOT chamber that has a flat octagonal shape with a height
of 46mm. This allows close placement of magnetic coils to the
atomic clouds for efficient operation of the magnetic transport.
The octagonal chamber has indium-sealed quartz windows with a
broadband antireflection (AR) coating, one small window for optical
pumping and six larger ones with a clear diameter of 40mm that
allow large MOT beams. In extension of the oven chamber, a second
five-way connector is attached to the MOT chamber. To one end a
window for the Zeeman slower beam is connected and is heated up to
165 °C to prevent coating of lithium atoms. Furthermore a 50 l/s
ion pump (same type as above) guarantees low pressure in the MOT
chamber and an ion gauge (Varian, UHV-24p) monitors the vacuum
pressure. Finally, a six-channel electrical feedthrough (VTS
Schwarz), which is connected from the bottom, is used for six
atomic dispensers.
The MOT chamber is then connected to UHV chamber via two
differential pumping tubes of 10 cm and 7.4 cm length and with an
inner diameter of 8mm. The tubes are intersected by a pneumatically
actuated valve that allows opening the vacuum of the MOT chamber
without affecting the vacuum in the UHV chamber or vice versa. Low
pressure is achieved by a large ion pump, 125 l/s (Varian, VacIon
Plus 150 StarCell), and a titanium sublimation pump (Thermionics)
that increases the pumping speed for reactive, getterable gases
like hydrogen and nitrogen. An additional ion gauge (same type as
above) measures the vacuum pressure and is switched off during the
experiment. In extension from the MOT chamber to the tubes a second
window for optical pumping and detection is connected to the end of
the UHV chamber. Perpendicular to this extension line a glass cell
is attached to the chamber that allows, in combination with an
additional window on the other side of the glass cell, excellent
optical access along six axes. The glass cell is a quartz glass by
Helma with a broadband AR coating (R < 0.5% for 512 − 1064 nm at
normal incidence). It consists of two parts, a rectangular part
with outer dimensions of 26×26×70.5mm3 and a wall thickness of 4mm
and a circular part of two stacked circular discs. The discs have
an outer diameter of 37mm and 50mm, a length of 19mm and a central
hole of 18mm. The glass cell is mounted to the steel chamber with a
spring loaded sealing ring (Garlock, HNV 200 Helicoflex Delta)
in-between.
After installing the vacuum system, all chambers are pumped down by
an attached turbo- molecular and membrane pump. To allow outgassing
from the bulk material, the system is baked out at roughly 200 °C
for several days. During bake-out, the atomic vapor dispensers and
the filaments of the titanium sublimation pump are initialized.
After gradually reducing the temperature, the ion pumps are
switched on and a pressure of a few times 10−10 mbar is reached in
the MOT chamber. The pressure in the UHV chamber is below the
detection limit of 1 × 10−11 mbar.
27
3. Experimental Setup
3.3 Atomic sources
Due to different physical and chemical properties of 6Li, 40K and
87Rb different strategies are used for loading the MOT. The species
potassium and rubidium are loaded from background gas, provided by
atomic vapor dispensers. For lithium this experimental simple
loading scheme would be inefficient due to a much lower saturation
pressure (see also Voigt (2004)) and a smaller mass resulting in a
small fraction of atoms travelling at speeds below the MOT capture
velocity. Therefore, lithium is loaded by a Zeeman slower, which
requires extra magnetic coils and additional lasers.
3.3.1 Dispensers for Rb and K
40K and 87Rb are loaded from background gas into the MOT. The atoms
are emitted from electrically heated dispensers through a redox
reaction between a salt and a reducing agent. In total, three
dispensers for each species are installed in the vacuum. The
dispensers are mounted on two macor rings and placed closed to the
MOT in hollows of the octagonal MOT chamber. Direct emission into
the MOT is blocked by an additional shielding wire. The electrical
connection of the dispensers allows to run one or two dispensers
for each species. In the experiment, usually only two dispensers of
each species are used to have one as a backup.
Due to a natural abundance of 28% for 87Rb, commercial dispensers
(SAES Getters, Rb/NF/7/25FT10+10) are used for loading 87Rb into
the MOT. For 40K the natural abun- dance is only 0.0117 (1) % and
does not allow the application of commercial dispensers.
Therefore, dispensers have to be build using enriched potassium
(DeMarco et al., 1999b), which relies on the following redox
reaction:
2KCl + Ca → 2K + CaCl2 (3.1)
In the first dispenser generation a 40K abundance of 3% is used. A
detailed description of the extensive manufacturing process can be
found in (Henkel, 2005).
For an advanced second generation of 40K dispensers the production
process is improved and a larger abundance of 6% is used (MaTeck,
Julich). In this version, 10mg of KCl and 20mg of pure calcium
(Sigma-Aldrich, pureness > 99.99%) are filled in three
containers made of a nickel-chromium foil (Goodfellow Inc.,
Ni80/Cr20). For this 2nd generation, the whole production process
is done under a protective atmosphere with dry argon. Furthermore,
to significantly suppress impurities, all components are baked-out
in a test vacuum setup before installing them in the MOT chamber.
In addition, an improved shielding of the MOT region with thicker
wires is installed.
3.3.2 Zeeman slower
Lithium is loaded into the MOT with the Zeeman slower technique,
which was first demon- strated by (Phillips and Metcalf, 1982). The
principle of this elegant and efficient technique relies on a
combination of laser cooling (Hansch and Schawlow, 1975) and the
Zeeman effect. An atomic beam is slowed down and cooled by the
light pressure of a resonant counter- propagating laser beam. To
keep the atoms always in resonance with the laser light, a
position-dependent magnetic field along the atomic beam compensates
the Doppler shift with the Zeeman shift.
An detailed description of our Zeeman slower design can be found in
(Taglieber, 2008).
28
3.4 Laser systems
3.4 Laser systems
Cold atom experiments require special laser systems for trapping,
detecting and manipula- tion. Working with three different species
simultaneously is a challenging aspect on the laser systems. In the
final stage of the experiment with Feshbach resonances, 4 laser
systems with 14 lasers and 8 laser locks are used. All of them have
to stay in lock for several hours and laser power stability is a
crucial aspect to guarantee constant atom numbers. For the build-up
spe- cial care is taken on stability and on reliability. All laser
systems are continuously improved during handling the
experiment.
In the following, I will first describe the energy levels and then
the laser systems of the three species. Finally, I show the setup
for optically trapping the atoms in a far detuned optical dipole
trap (ODT).
3.4.1 Energy levels
This section briefly describes specific properties of 87Rb, 40K and
6Li, the relevant energy levels and the corresponding transitions
for trapping, detecting and manipulation.
The well-known 87Rb atom is magneto-optically trapped and laser
cooled on the D2-line
2S1/2, F = 2 ⟩
⟩
⟩
transition.
For the fermionic isotope 40K the trapping transition is between
the lowest lying levels
4S1/2, F = 9/2 ⟩
4S1/2, F = 7/2 ⟩
⟩
and requires higher laser intensity than in the rubidium case due
to the comparatively smaller branching ratio between the excited
states. Imaging can be done on closed transitions at variable
magnetic fields. A special issue of 40K is the inverted hyperfine
structure due to a positive nuclear g-factor. For stability
arguments in atomic mixtures this is an important aspect (see Sec.
5.1.2).
In 6Li the hyperfine level structure of the excited
2P1/2
⟩
2S1/2, F = 3/2 ⟩
⟩
on the D2-line (671 nm) have comparable population rates and both
of them need balanced laser intensities for MOT operation. Another
important result is the inefficiency of polarization gradient
cooling. Moreover, to have a dark state, optical pumping is carried
out on the D1-line
2S1/2, F = 3/2 ⟩
⟩
and not on the D2-line. Imaging, as in the potassium case, is
possible at variable magnetic fields. For low field imaging,
additional repumping light pumps the atoms back into the cycling
transition.
3.4.2 Rubidium
The rubidium system is based on three external cavity diode lasers
(ECDL) (Wieman and Hollberg, 1991; Ricci et al., 1995) that are
frequency stabilized by either saturated absorp- tion
frequency-modulation (FM) spectroscopy (Bjorklund et al., 1983;
Drever et al., 1983), by a Doppler-free dichroic lock (DFDL) (Wasik
et al., 2002) or by a beat lock technique
29
F=3/2
F=1/2
sl o
w er
tr ap
p in
10 Ghz
O P
6 Li
40 K
87 Rb
Figure 3.2: Atomic energies levels and transitions that are used to
trap and detect rubidium, potassium and lithium.
30
TA
imaging
(Schunemann et al., 1999).
Laser ECDL1 provides the necessary repumper light for trapping and
optical pumping. Laser ECDL2 gives the light for imaging and
optical pumping. Laser light from ECDL3 is amplified by an
commercial tapered amplifier (TA) system (Toptica, TA-780) and
gives 280mW of trapping light after an optical fiber. The light
intensity is controlled by an electro- optic modulator (EOM)
(Gsanger, LM 0202). Acousto-optic modulators (AOM) are used to
shift laser light frequency in the desired way and to switch off
the light that goes into optical fibers for cleaning. Residual
light is blocked by mechanical shutters. Further details can be
found in (Taglieber, 2008).
3.4.3 Potassium
The laser system for potassium uses two ECDL and two slave lasers
(SL). These lasers are composed of AR-coated laser diodes
(Eagleyard, EYP-RWE-0790) with high output powers (ECDL 20mW, SL
40mW). The coating allows to tune the wavelength easily from the
free-running wavelength at 790 nm to 767 nm.
⟩
⟩
of the D2-line in 39K. To improve the lock stability, the
spectroscopy cell is temperature stabilized additionally to 45.0°C.
This increases the error
31
Figure 3.5: Schematic design of the 6Li laser system.
signal of the locking signal by more than a factor of three.
Laser light from ECDL1 is divided in two paths that seeds two slave
lasers SL1 and SL2 for light amplification. A double and a
quadruple pass AOM controls the frequency. Both beams are then
co-aligned on a beam splitter. One beam is then coupled into a
fiber for optical pumping and imaging. The other one is amplified
by an self-made TA booster system (TA chip by Eagleyard,
EYP-TPA-0765-01500-3006-CMT03) for trapping and repumping. After an
optical fiber for cleaning the spectral mode, typically 350mW are
achieved for the trapping transition.
Finally, an extra ECDL2 is referenced on ECDL1 by an offset lock
and provides light for high-field imaging. The offset frequency is
given by an computer controlled signal generator (R&S SML
02).
3.4.4 Lithium
A first version of the lithium laser system can be found in my
diploma thesis (Voigt, 2004) and is significantly improved and
extended for the measurements discussed in Cha. 5 and 6.
The old laser diodes (Panasonic, LNCQ 05 PS), which are used for
the measurements discussed in Cha. 4, are exchanged by laser diodes
with a much higher laser output power (Mitsubishi, ML101J27). They
are specified with an output power of 120mW and a free- running
wavelength of 661 nm at 25 °C. To run them at 671 nm the laser
diodes are heated up to more than 70 °C and the currents are 250mA
for the ECDL and 310mA for the SL, well above specification. A test
laser, ran continuously with these settings for more than half a
year, showed no degradation. For the slave lasers stable seeding
has been obtained for seeding powers between 100µW and a few mW.
Typical output powers for the slave lasers after the isolator are
80mW.
The larger available laser power also allows to change some optical
paths. The frequency lock of ECDL1 is now optically decoupled from
the first AOM to allow fast frequency changes. Moreover, an extra
AOM is added in front of the trapping and repumper fiber to control
the laser light intensity. These two last changes are important for
a reliable operation of the lithium compressed MOT stage (Petrich
et al., 1994) of the experimental cycle (see Sec. 5.1.1).
With the larger trapping and slower power the atom number of the
Li-MOT is increased
32
3.4 Laser systems
by more than two orders of magnitude than in the previous
case.
For optical pumping it is beneficial to optically pump the atoms
into a dark state. The only transition with an optically resolvable
dark state is the D1-line. Therefore, a third ECDL3, running on the
D1-line, is installed in the existing setup. The frequency is
stabilized with an FM-lock using the same spectroscopy cell as
ECDL1.
3.4.5 Optical system for MOT, detection and optical pumping
The vacuum chamber and the laser systems, described above, are
installed on two different optical tables. For MOT operation,
optical pumping and detection laser light is coupled into
polarization maintaining optical fibers to guide the light to the
desired position. The fibers also clean the spatial profile of the
beams and increases the beam quality, which is important for a
robust MOT and for high quality absorption images.
Simultaneous trapping of all three species in a MOT requires to
superimpose the individual laser beams for trapping and repumping
on six MOT beams, four in the horizontal and two in the vertical
plane. In contrast to lithium and potassium, repumping light for
rubidium is only in the horizontal beams. The trapping power for
87Rb and 40K is equal in all six beams, respectively. Trapping and
repumper light intensity for 6Li is comparable within each beam and
the vertical beams have two times more power than the horizontal
ones. A detailed description of the optical setup is given in (Tagl